We study the tasks of collective coin flipping and leader election in the full-information model. We prove new lower bounds for coin flipping protocols, implying lower bounds for leader election protocols. We show that any $k$-round coin flipping protocol, where each of $\ell$ players sends 1 bit per round, can be biased by $O(\ell/\log^{(k)}(\ell))$ bad players. For all $k>1$ this strengthens previous lower bounds [RSZ, SICOMP 2002], which ruled out protocols resilient to adversaries controlling $O(\ell/\log^{(2k-1)}(\ell))$ players. Consequently, we establish that any protocol tolerating a linear fraction of corrupt players, with only 1 bit per round, must run for at least $\log^*\ell-O(1)$ rounds, improving on the prior best lower bound of $\frac12 \log^*\ell-\log^*\log^*\ell$. This lower bound matches the number of rounds, $\log^*\ell$, taken by the current best coin flipping protocols from [RZ, JCSS 2001], [F, FOCS 1999] that can handle a linear sized coalition of bad players, but with players sending unlimited bits per round. We also derive lower bounds for protocols allowing multi-bit messages per round. Our results show that the protocols from [RZ, JCSS 2001], [F, FOCS 1999] that handle a linear number of corrupt players are almost optimal in terms of round complexity and communication per player in a round. A key technical ingredient in proving our lower bounds is a new result regarding biasing most functions from a family of functions using a common set of bad players and a small specialized set of bad players specific to each function that is biased. We give improved constant-round coin flipping protocols in the setting that each player can send 1 bit per round. For two rounds, our protocol can handle $O(\ell/(\log\ell)(\log\log\ell)^2)$ sized coalition of bad players; better than the best one-round protocol by [AL, Combinatorica 1993] in this setting.

翻译：本研究在全信息模型中探讨集体掷币与领导者选举任务。我们证明了掷币协议的新下界，从而推导出领导者选举协议的下界。我们证明：任何$k$轮掷币协议（其中$\ell$位参与者每轮发送1比特信息）均可被$O(\ell/\log^{(k)}(\ell))$个恶意参与者操纵。对于所有$k>1$的情况，该结果强化了先前下界[RSZ, SICOMP 2002]——原下界排除了能抵抗$O(\ell/\log^{(2k-1)}(\ell))$个恶意参与者的协议。由此我们确立：任何容忍线性比例腐败参与者且每轮仅传输1比特的协议，必须运行至少$\log^*\ell-O(1)$轮，改进了先前最佳下界$\frac12 \log^*\ell-\log^*\log^*\ell$。该下界与当前最佳掷币协议[RZ, JCSS 2001], [F, FOCS 1999]的轮数$\log^*\ell$相匹配——这些协议能处理线性规模的恶意联盟，但要求参与者每轮发送无限比特信息。我们还推导了允许每轮发送多比特信息的协议下界。结果表明，[RZ, JCSS 2001], [F, FOCS 1999]中处理线性数量腐败参与者的协议在轮复杂度和每轮通信量方面近乎最优。证明下界的关键技术要素是一项新结果：通过共用恶意参与者集合与针对每个被操纵函数的专用小型恶意集合，可操纵函数族中的大多数函数。我们在每参与者每轮发送1比特的设置中给出了改进的常数轮掷币协议。对于两轮协议，我们的方案可处理规模为$O(\ell/(\log\ell)(\log\log\ell)^2)$的恶意联盟，优于该设置下[AL, Combinatorica 1993]的最佳单轮协议。